保罗·古尔丁Paul Guldin

保罗·古尔丁被父母命名为哈巴谷古尔丁。尽管有犹太血统，他的父母是新教徒，他们以这种信仰抚养古尔丁。学徒期满后，他成为一名金匠，并在青少年时期从事这一行业，辗转于不同的德国城市。在1590年代后半期，他在弗赖辛工作，在那里他读了许多书，使他对所信奉的新教产生了怀疑。他去了弗赖辛的魏恩斯特凡本笃会修道院，向修道院的院长解释了他的疑虑。这是一个艰难的决定，但他接受了院长的建议，放弃了他从小被抚养的新教。此时，他将名字从哈巴谷（一个来自十二小先知之一的犹太名字）改为古尔丁，因为他将古尔丁视为将基督教带给外邦人的犹太人。古尔丁在20岁时皈依天主教，并在慕尼黑加入耶稣会，成为助理修士。在此之前，古尔丁肯定没有学过数学，事实上，除了能读会写之外，他是否接受过多少教育都值得怀疑。然而，耶稣会是一个致力于严格教育的修会，古尔丁经历了漫长的教育过程，最终获得神学博士学位。几年后，他成为耶稣会学者，再后来，他被任命为耶稣会神父。

古尔丁展现出相当高的数学才能，因此1609年被送往罗马的耶稣会罗马学院，师从当时任数学教授的克里斯托佛·克拉乌。尽管克里斯托佛·克拉乌不以数学发现闻名，但他是一位极其出色的教师，古尔丁从他的讲座中获得了深刻的数学理解。然而，克里斯托佛·克拉乌是一位古典数学家，只教授欧几里得的几何方法，古尔丁也采取这种古典进路，反对当时开始出现的微积分新思想。在受克里斯托佛·克拉乌指导之后，古尔丁在罗马的耶稣会学院教授数学。随后，1617年他迁往格拉茨的耶稣会学院，但几年后严重的健康问题迫使他放弃授课。事实上，他的第一部著作在他抵达格拉茨后不久就出版了。在Refutatio elenchi calendarii Gregoriani a Setho Calvisio conscriptiⓉ（对Sethus Calvisius所写格里高利历清单的反驳）（1618）中，古尔丁为他的老师克里斯托佛·克拉乌的历法改革提议辩护。我们知道古尔丁此时对数学的看法，因为他后来发表了他1622年讲授的一堂课[2]：-

[古尔丁]按照亚里士多德的分类定义数学，将其视为介于物理学与形而上学之间的哲学部分。数学是从可感事物中抽象出量的科学。关于纯数学，算术被描述为离散量的科学，几何学被描述为连续量的科学。值得注意的是，代数被加入纯数学的传统学科之中；当然，古尔丁的代数观念在很大程度上依赖于弗朗索瓦·韦达的工作。

同样在1622年，他出版了一部关于地球重心的著作。他接受了这样的观点：每个大物体的重心都试图移动，以与宇宙的重心重合。一个有趣的推论是，古尔丁论证地球会不断运动。

1623年，他被派往维也纳，被任命为大学数学教授。1629年，耶稣会派他去西里西亚萨甘公国的耶稣会文理中学任教，该机构由阿尔布雷希特·华伦斯坦在1627年被封为萨甘亲王后建立。在那里任教一段时间后，古尔丁回到维也纳的教授职位，一直待到1637年返回格拉茨。

古尔丁参与的一段有趣通信是与约翰内斯·开普勒的通信。遗憾的是，只有约翰内斯·开普勒写给古尔丁的信被保存下来，但尽管如此，它们仍给我们提供了有趣的信息。约翰内斯·开普勒在1618年至1628年间写了十一封信给古尔丁，这些信在[15]中得到了讨论。约翰内斯·开普勒在科学事务和宗教事务上都寻求古尔丁的建议，他还请古尔丁利用其在宫廷中的影响力。[15]：-

约翰内斯·开普勒信任古尔丁的帮助并赞赏其建议的一个例子，见于开普勒1624年8月30日的信。他曾寄给古尔丁一份请愿书，要转呈皇帝斐迪南二世（1578-1637），以推动《鲁道夫星表》的出版。开普勒现在问他，为了推进他的事务，该带什么去维也纳。这一事实也表明，古尔丁在维也纳宫廷是一位非常有影响力的人物。……开普勒写给古尔丁的最后两封信表达了他对古尔丁期望开普勒可能皈依天主教会的担忧。

在他们通信的整个时期，约翰内斯·开普勒的经济状况都很差，古尔丁担心约翰内斯·开普勒买不起望远镜来开展科学工作。古尔丁的一位耶稣会朋友Nicolas Zucchi是望远镜制造者，古尔丁请他送一架望远镜给约翰内斯·开普勒。约翰内斯·开普勒回复古尔丁，表明他对这份礼物极为感激，并寄给古尔丁他的书，详细记述了他用它做出的发现：-

致最可敬的神父古尔丁，耶稣会司铎，可敬而博学的人，亲爱的庇护者。此时几乎没有人比您更适合与我讨论天文学问题了……因此，由您修会中在此的成员转达的您的问候，更令我感到愉快……我认为您应当从我这里收到我因使用您的礼物（望远镜）而获得喜悦的第一份文学成果。

古尔丁最重要的著作是Centrobaryca seu de centro gravitatis trium specierum quantitatis continuae Ⓣ（论三种图形的重心），于1635年至1641年间分4卷出版。第1卷以他1622年所作的描述数学科学的演讲开篇（上文已讨论）。由于整部著作都关于重心，值得给出古尔丁在这一卷中的定义：-

任何有限量的重心是这样一个点，它位于该量内部、或其边界上、或其外部，围绕它各侧都有相等矩的部分。因为无论是重心本身，还是通过重心任意画出的直线或平面，总会把所给图形切成重量相等的部分。

特别是在第1卷中，他讨论了地球的重心。第2卷出版于1640年，包含了他如今著名的重心规则。这本书的主要目的是研究由其他图形旋转得到的图形，例如由半圆绕其直径旋转得到的球。以下是古尔丁定义的一个例子：-

旋转是一种简单而完美的圆周运动，围绕一个固定中心或一条不动的轴，称为“旋转轴”，绕着一个点、一条线或一个平面转动，几乎像留下痕迹一样，描绘或生成一个圆形的量，或是一条线，或是一个面，或是一个体。

他的著名规则可以表述为以下形式：-

如果一个平面图形绕其平面内的轴旋转，则所形成的立体体积等于该图形的面积与重心所经过的距离的乘积。

古尔丁 尤其不幸，因为他在这一结果上被指控剽窃。历史学家 大卫·尤金·史密斯 毫不含糊地指出了这一点 [14]：-

古尔丁 定理的精髓出现在 帕普斯 的著作中。这些著作出版于 1588 年、1589 年和 1602 年，比 古尔丁 出版（1641 年）他的《Centrobaryca》早了大约一代人……当时的学者们熟知这些著作，认为它们构成了希腊晚期最重要的几何学著作。像 古尔丁 这样的人竟然不知道这样一部著名著作中的这一重要陈述，是完全不可想象的。

历史学家们花了一些时间争论这一点，不幸的是常常使用错误的信息。最公正的评价无疑是 Ivor Bulmer-Thomas 的 [5]：-

并不是说 古尔丁 是一个怀有恶意的人。《De centro gravitatis》的证据表明，他热切地给予应得的荣誉。……不仅有对 费德里哥·哥曼迪诺 本人的友好引用，还有对 克里斯托佛·克拉乌、约翰·缪勒、Orontius、Freigius、彼得吕斯·拉米斯 和 吉罗拉莫·卡尔达诺 的友好引用。如果 古尔丁 意识到对 帕普斯 有任何亏欠，他肯定会承认。这并不排除一种他并未意识到的亏欠。很可能 帕普斯 的表述在许多年前被读到后，已经沉入他的潜意识，而当他 1640 年开始写作时，他真诚地相信自己正在创作原创的东西。……根据所有证据，似乎最好对 古尔丁 作有利推定——宣判他没有有意识的剽窃，但接受他可能从潜意识中产生了他遥远阅读的成果。

第 3 卷包含关于圆锥、圆柱和旋转体的表面积和体积的著作。古尔丁 利用第 4 卷攻击其他数学家所使用的方法。以这种方式被攻击的一位数学家是 博纳文图拉·卡瓦列里 [2]：-

博纳文图拉·卡瓦列里与古尔丁之间的争论通常与古尔丁对博纳文图拉·卡瓦列里使用不可分量的反对意见联系在一起被提及。尽管这可能是博纳文图拉·卡瓦列里与古尔丁之间的主要问题，但更仔细地阅读这场争论将使我们能够指出其他有趣问题的存在……

争论实际上围绕这样一个事实：古尔丁是一位遵循古希腊数学家方法的古典几何学家。然而，他的第一点是指责博纳文图拉·卡瓦列里抄袭了约翰内斯·开普勒的Stereometria DoliorumⓉ（《酒桶立体测量学》）（1615年）和Sover的Curvi ac Recti ProportioⓉ（《曲线与正比例》）（1630年）。他的论点中有些内容与约翰内斯·开普勒有关，因为在那部著作中约翰内斯·开普勒确实将圆视为由无穷小组成的无限多边形。然而，博纳文图拉·卡瓦列里的不可分量与约翰内斯·开普勒的无穷小是不同的。至于提到Sover，博纳文图拉·卡瓦列里在辩护中指出，他的书写于Sover的书出版之前。古尔丁攻击博纳文图拉·卡瓦列里的不可分量，论证说当一条线绕轴旋转生成一个曲面时，该曲面不仅仅是一组线。他写道（见[2]或[13]）：-

在我看来，没有几何学家会同意博纳文图拉·卡瓦列里所说的曲面是、并且可以用几何语言称为“这样一个图形的所有线”；事实上，若干条线，或所有的线，从来都不能被称为曲面；因为线的数量无论多么大，都不能构成哪怕最小的曲面。

正如Mancosu所写[2]：-

古尔丁是一位“古典主义”几何学家，浸淫于明确构造的观念，对几何学领域中的无穷概念持怀疑态度，并警惕最终陷入连续统的原子论理论的风险。

如果问古尔丁和博纳文图拉·卡瓦列里谁是对的，那么答案必定是博纳文图拉·卡瓦列里。然而，这并不使古尔丁的工作变得无用，因为在他对古典方法的支持中，他迫使像博纳文图拉·卡瓦列里这样的创新者更深入地思考并更严格地证明他们的方法。

为了理解为什么古尔丁在第四卷中攻击弗洛伦斯·南丁格尔·大卫 Rivaltus，我们必须认识到，遵循其老师克里斯托佛·克拉乌所采取的路线，古尔丁不相信阿基米德的结果得到了令人满意的证明。阿基米德的证明曾遭到人文主义者Joseph Scaliger在其论著Cyclometrica Ⓣ（《圆的测量》）（1594年）中的强烈攻击。大卫 Rivaltus于1615年以同样强有力的方式为阿基米德辩护，回应了这些批评。古尔丁则反过来攻击Rivaltus，写道[13]：-

确实，直接证明总是赢得掌声并战胜否定性证明以及那些归谬或归为不可能的证明，无论Rivaltus……说什么。

事实上，在1641年出版的第4卷中，古尔丁试图重建当时所认为的大部分数学。他的目标是不使用矛盾法来证明结果。例如，他写道（见[2]或[13]）：-

因此，我们将在第四卷中，通过我们来自旋转和源自重心的原理，直接证明阿基米德在《球与圆柱》以及类似地在《锥体与椭球体》中所证明的主要命题，而他本人是通过矛盾法确立这些命题的。

这里有一点值得注意。尽管古尔丁和博纳文图拉·卡瓦列里之间存在争论，但两人都同意矛盾证明是不可取的。也许令人惊讶的是，博纳文图拉·卡瓦列里为其方法辩护，论证说他使用无穷小量使他能够不使用“矛盾证明”来证明阿基米德的结果。

最后，让我们看看古尔丁至今仍存在的藏书的一些细节。首先要注意的事实是，古尔丁被允许保留自己的藏书，这对耶稣会内部的人来说是非常不寻常的。大约300卷已被确认属于古尔丁的藏书，尽管许多后来重新装订的其他书籍可能原本属于该藏书，但在重新装订中失去了证据。大多数书籍似乎没有先前的所有者，因此古尔丁是在购买新书。由于有三本书被标记为属于该藏书，却是在他去世后购买的，可以推断他有一名图书管理员来照管他的藏书。古尔丁对哪些学科感兴趣？当然有伽利略的著作，算术书籍如奥文斯·菲内的，阿方索星表，以及费德里哥·哥曼迪诺的著作。古尔丁显然对力学非常感兴趣，因为他有一整套关于这个主题的书籍，包括关于军事装备、防御工事、火炮和烟火术的。建筑学显然也让他感兴趣，因为他也有几部关于该主题的著作。

Paul Guldin was named Habakkuk Guldin by his parents. Although of Jewish descent, his parents were Protestants and they brought Guldin up in that faith. He became a goldsmith, after serving an apprenticeship, and worked at that trade during his teens moving between different German cities. In the second half of the 1590s, he was working in Freising and there he read a number of books which led him to have doubts about the Protestant religion he was practicing. He went to the Benedictine abbey of Weihenstephan, in Freising, and explained his doubts to the prior in the abbey. It was a hard decision, but he took the advice of the prior and renounced the Protestant religion in which he had been brought up. At this point he changed his name from Habakkuk (a Jewish name coming from one of the twelve minor Prophets) to Paul since he saw Paul as the Jew who took Christianity to the Gentiles. Guldin became a convert to Catholicism at the age of 20 and joined the Jesuit Order in Munich as a Coadjutor Brother. Up to this point, Guldin would certainly not have studied mathematics, in fact it is doubtful if he had received much education beyond being able to read and write. The Jesuits, however, were an Order committed to rigorous education and Guldin went through the lengthy educational process leading to a doctorate in divinity. After a few years he became a Jesuit Scholastic and, still later, he was ordained a Jesuit priest.

Since Guldin showed considerable mathematical abilities so, in 1609, he was sent to the Jesuit Collegio Romano in Rome to study under Clavius who was the professor of mathematics there. Although not known for mathematical discoveries, nevertheless Clavius was an exceptionally good teacher and Guldin gained deep mathematical understanding from his lectures. Clavius was, however, a classical mathematician teaching only Euclid's geometric methods and Guldin would also take this classical approach and oppose the newer ideas of the calculus which were beginning to appear around this time. After being instructed by Clavius, Guldin taught mathematics at the Jesuit College in Rome. Then, in 1617, he moved to the Jesuit College in Graz but after a few years a severe health problem forced him to give up lecturing. In fact his first work was published shortly after he arrived in Graz. In Refutatio elenchi calendarii Gregoriani a Setho Calvisio conscripti Ⓣ (1618), Guldin defended his teacher Clavius's proposals for calendar reform. We know Guldin's views on mathematics at this time since he later published a lecture which he had delivered in 1622 [2]:-

[Guldin] defines mathematics according to the Aristotelian classification, as that part of philosophy lying between physics and metaphysics. Mathematics is the science that considers quantity abstracted from sensible matter. Concerning pure mathematics, arithmetic is described as the science of discrete quantity, and geometry as the science of continuous quantity. It is interesting to note the addition of algebra to the traditional disciplines of pure mathematics; of course, Guldin's conception of algebra depends in large part on the work of Viète.

Also in 1622 he published a work on the centre of gravity of the Earth. He accepted the view that the centre of gravity of every large body tries to move so that will coincide with the centre of gravity of the universe. An interesting consequence was that Guldin argued that the Earth would constantly be moving.

He was sent to Vienna in 1623 where he was appointed professor of mathematics at the University. In 1629 he was sent by the Jesuit Order to teach at the Jesuit Gymnasium in the Silesian principality of Sagan which had been established by Albrecht Wallenstein after he was made Prince of Sagan in 1627. After teaching there for some time, Guldin returned to his professorship in Vienna where he remained until 1637 when he returned to Graz.

One interesting correspondence which Guldin entered into was with Johannes Kepler. Unfortunately only Kepler's letters to Guldin have been preserved but, nevertheless, they give us interesting information. Kepler wrote eleven letters to Guldin between 1618 and 1628 and these are discussed in [15]. Kepler sought Guldin's advice both on scientific matters and on religious matters, and he also asks Guldin to use his influence in the court. [15]:-

[An] example of Kepler's trust in Guldin's help and the appreciation of his advice is shown in Kepler's letter from August the 30th 1624. He had sent Guldin a petition to be forwarded to emperor Ferdinand II (1578-1637) to promote the publication of the Rudolphinian Tables. Kepler asked him now what to bring with him to Vienna to help the progress of his affair. This fact also illustrates that Guldin was a very influential person at the imperial court in Vienna. ... Kepler's last two letters to Guldin express his uneasiness concerning Guldin's expectation of Kepler's possible conversion to the Catholic Church.

Kepler's financial position was poor throughout the period of their correspondence and Guldin was concerned that Kepler could not afford a telescope to carry out scientific work. One of Guldin's Jesuit friends, Nicolas Zucchi, was a telescope maker and Guldin asked him to give Kepler one of his telescopes. Kepler replied to Guldin showing that he was extremely grateful for the gift and sent Guldin his book detailing the discoveries he had made with it:-

To the very reverend Father Paul Guldin, priest of the Society of Jesus, venerable and learned man, beloved patron. There is hardly anyone at this time with whom I would rather discuss matters of astronomy than with you ... Even more of a pleasure to me, therefore, was the greeting from your reverence which was delivered to me by members of your Order who are here ... I think you should receive from me the first literary fruit of the joy that I have gained from using your gift (the telescope).

Guldin's most important work is Centrobaryca seu de centro gravitatis trium specierum quantitatis continuae Ⓣ published in 4 volumes between 1635 and 1641. Volume 1 begins with the lecture describing the mathematical sciences which he gave in 1622 (discussed above). Since the whole work is on centres of gravity it is worth giving Guldin's definition from this volume:-

The centre of gravity of any finite quantity is that point placed either inside that quantity, or on its boundary, or outside it, around which on all sides are parts of equal moment. For either the centre itself, or the straight line, or a plane however drawn through the centre, will always cut the proposed figure in parts of equal weight.

In particular, in Volume 1, he discusses the centre of gravity of the Earth. Volume 2 was published in 1640 and contains his, now famous, rule on centres of gravity. The main aim of this book was to study figures obtained by rotating other figures, for example the sphere obtained by rotating a semicircle about its diameter. Here is an example of Guldin's definition:-

A rotation is a simple and perfectly circular motion, around a fixed centre, or an unmoved axis, which is called the 'axis of rotation', turning around either a point, or a line, or a plane surface, which, almost as leaving a trace behind it, describes or generates a circular quantity, either a line, or a surface, or a body.

His famous rule can be stated in the following form:-

If a plane figure is rotated about an axis in its plane then the volume of the solid body formed is equal to the product of the area with the distance travelled by the centre of gravity.

Now Guldin has been particularly unfortunate in that he has been accused of plagiarism over this result. The historian David Smith states this in no uncertain terms [14]:-

The essence of Guldin's Theorem appears in the works of Pappus. These were published in 1588, 1589 and 1602, a generation or so before Guldin published (1641) his 'Centrobaryca' ... They were well known to scholars of that time as constituting the most important geometric work of the late Greek period. That a man like Guldin should have failed to know this important statement in such a well-known work is quite inconceivable.

Historians have spent some time arguing this point, often unfortunately using fallacious information. The fairest assessment is certainly that of Ivor Bulmer-Thomas [5]:-

It is not as though Guldin was a spiteful man. The evidence of 'De centro gravitatis' is that he was zealous to give credit where credit was due. ... there are friendly references not only to Commandinus himself but to Clavius, Regiomontanus, Orontius, Freigius, Peter Ramus, and Cardan. If Guldin had been conscious of any debt to Pappus, he would surely have acknowledged it. This does not rule out a debt of which he was unconscious. It may very well be that Pappus's enunciation, read many years earlier, had sunk into his unconscious mind, and that when he came to write in 1640, he genuinely believed that he was producing something original. ... In the light of all the evidence, it seems best to give Guldin the benefit of the doubt - to acquit him of conscious plagiarism, but to accept that he may have brought forth from his subconscious mind the fruit of his distant reading.

Volume 3 contains work on the surface and volumes of cones, cylinders and solids of revolution. Guldin uses Volume 4 to attack other mathematicians for the methods they are using. One mathematician to be attacked in this way is Cavalieri [2]:-

The debate between Cavalieri and Guldin is usually mentioned in connection with the objections made by Guldin to Cavalieri's use of indivisibles. Although that is probably the main issue between Cavalieri and Guldin, a more careful reading of the debate will allow us to indicate the existence of other interesting issues ...

The argument really centres around the fact that Guldin is a classical geometer following the methods of the ancient Greek mathematicians. His first point, however, is to accuse Cavalieri of plagiarising Kepler's Stereometria Doliorum Ⓣ (1615) and Sover's Curvi ac Recti Proportio Ⓣ (1630). There is something in his argument relating to Kepler since in that work Kepler does regard a circle as an infinite polygon composed of infinitesimals. However, Cavalieri's indivisibles are different from Kepler's infinitesimals. As to the reference to Sover, Cavalieri, in his defence, points out that he wrote his book before Sover's book was published. Guldin attacks Cavalieri's indivisibles by arguing that when a surface is generated by rotating a line about the axis, the surface is not just a set of lines. He writes (see [2] or [13]):-

In my opinion no geometer will grant Cavalieri that the surface is, and could, in geometrical language be called "all the lines of such a figure"; never in fact can several lines, or all the lines, be called surfaces; for, the multitude of lines, however great that might be, cannot compose even the smallest surface.

As Mancosu writes [2]:-

Guldin was a "classicist" geometer, steeped in the idea of explicit construction, sceptical of considerations of infinity in the domain of geometry, and wary of the risk of ending up with an atomistic theory of the continuum.

If one asks whether Guldin or Cavalieri is right, then the answer must be Cavalieri. However, this does not make Guldin's work useless for in his support of the classical approach, he forced innovators like Cavalieri to think more deeply and to justify their methods more rigorously.

In order to understand why Guldin attacks David Rivaltus in Volume 4 we must realise that, following the line taken by his teacher Clavius, Guldin does not believe that Archimedes' results are proved in a satisfactory manner. Archimedes' proofs had been strongly attacked by the humanist Joseph Scaliger in his treatise Cyclometrica Ⓣ (1594). David Rivaltus replied to these criticisms in 1615 with an equally strong defence of Archimedes. Guldin, in turn, attacks Rivaltus writing [13]:-

Indeed ostensive demonstrations have always had the applause and the victory over negatives and those reducing to absurdity or impossibility, whatever Rivaltus ... says.

In fact in Volume 4, published in 1641, Guldin attempts a reconstruction of most of what was then considered mathematics. His aim is to prove the results without using the method of contradiction. For example, he writes (see [2] or [13]):-

We will therefore prove ostensively, in this fourth book, through our principles from rotation and originating from the centre of gravity, the main propositions proved by Archimedes on the 'Sphere and the Cylinder' and similarly on 'Conoids and Spheroids', which he himself had established by contradiction.

One point here is worth noting. Despite the argument between Guldin and Cavalieri, both agreed that proofs by contradiction are undesirable. Perhaps surprisingly, Cavalieri justified his approach by arguing that his use of infinitesimals allowed him to prove Archimedes' results without the use of 'proof by contradiction'.

Finally, let us look at some details of Guldin's library which still exists. The first fact to note is that Guldin was allowed to keep his own library, something which was very unusual for those within the Jesuit Order. Around 300 volumes have been identified as belonging to Guldin's library although many other books which have been subsequently rebound may have originally belonged to the library but have lost the evidence in the rebinding. Most of the books appear to have had no previous owners and so Guldin was buying new books. Since three books are marked as belonging to the library, yet were purchased after his death, one can deduce that he had a librarian to look after his library. What subjects interested Guldin? There are of course the works of Galileo, books on arithmetic such as that by Oronce Finé, the Alphonsine Tables, and works by Frederico Commandino. Guldin was clearly very interested in mechanics because he had a whole series of books on this subject including ones on military equipment, fortifications, artillery and pyrotechnics. Architecture also clearly interested him since he also had several works on that topic.